Define ϕ≡φ−a/cost, so that ϕ= 0 on the hypebola. Now the parameterization reads
x(u, t, ϕ) = ϕ(c−acosucost) +ac/cost−a
2cosu+b2cosu
Figure 4.6: Left: Complete Dupin cyclides of Type I FCD. Right: Type II FCD. Physical focal sets are shown as black curves. Virtual focal sets are shown as light gray curves.
= ϕ(c−acosucost) +ac/cost−c
2cosu
a−ccosucost y(u, t, ϕ) = − bϕsinucost
a−ccosucost
z(u, t, ϕ) = bsint(ϕ−ccosu) +abtant
a−ccosucost (4.5)
The intervals for ϕare [ccosu−a/cost,0] for Type I FCDs and [0,∞) for Type II FCDs. ϕis nonpositive for Type I FCDs and nonnegative for Type II FCDs.
Now consider the following transformation: Send the ellipse major and minor axis lengths to their opposites and rotate the FCD about ˆz by π, while keeping the sign of
cunchanged. In symbols,
Then, parameterization (4.5) becomes
x(u, t, ϕ) = −ϕ(c−acosucost) +ac/cost−c
2cosu
a−ccosucost y(u, t, ϕ) = bϕsinucost
a−ccosucost
z(u, t, ϕ) = −bsint(−ϕ−ccosu) +abtant
a−ccosucost (4.7)
Comparing with parameterization (4.5), we see that transformation (4.6) is equivalent to taking ϕ→ −ϕ. (There is an additional reflection through the xy plane required to formally return to the original parameterization, but the FCD was symmetric under this reflection in the first place.) This means that taking a, b to −a,−b, along with a rotation by π, transforms an FCD of type I into an FCD of type II with the same eccentricity, and vice versa.
A continuous transformation ofaandb into their opposites can be accomplished by taking them through 0, at which point the smectic layer configuration passes through concentric spheres. To see this more clearly, we can rewrite parameterization (4.5) with numerator and denominator divided by a:
x(u, t, ϕ) = ϕ(e−cosucost) +ae/cost−ae
2cosu 1−ecosucost y(u, t, ϕ) = − √ 1−e2ϕsinucost 1−ecosucost z(u, t, ϕ) = √
1−e2(sint(ϕ−aecosu) +atant)
1−ecosucost (4.8) Taking a→0, x(u, t, ϕ) = ϕ(e−cosucost) 1−ecosucost y(u, t, ϕ) = − √ 1−e2ϕsinucost 1−ecosucost
z(u, t, ϕ) =
√
1−e2ϕsint
1−ecosucost (4.9)
which reduces tox2+y2+z2 =ϕ2. Alternatively, we could takea, bto−a,−b through
±∞. At a = ±∞, the FCD becomes a family of cylinders coaxial about the line
x=y= 0. This is less easy to see from the parameterization, but it is geometrically obvious for the e= 0 case where the focal sets are the line x= y= 0 and the circle in the xy plane at infinity; furthermore, the structure is invariant under Lorentz boosts and therefore does not change fore 6= 0. [4].
This relation between Type I and Type II FCDs under reversing the sign of a, b
is seen clearly in the general formula of Ref. [4] for zero-eccentricity FCDs as the “product of two cones”,
(s+r)2+z2−φ2 (s−r)2+z2−φ2
= 0 (4.10)
where s= p
x2 +y2. Setting the first factor equal to zero gives Type II FCDs; setting
the second factor equal to zero gives Type I FCDs. It is obvious that the two factors are related by reversing the sign of r.
4.1.4
The law of corresponding cones
The law of corresponding cones [40] allows multiple Type I FCDs to be joined together with no dislocations or discontinuities in the smectic layer normal. We now review the law of corresponding cones for a family of FCDs. Theith FCD is bounded by a right circular coneCi that has its apex P on the hyperbola Hi and that includes the ellipseEi. Thus, Ci consists entirely of generators, straight lines which are normal to the smectic layers and which each connect a point on Ei to a point on Hi. In the “fan” texture typical of kinetically trapped FCD assemblies, when two ellipsesE1 and
E2 are tangent at a point Q, then their boundary cones C1 and C2 are tangent along
an entire generator if C1 and C2 have a common apex P where the hyperbolae H1
and H2 intersect (Fig. 4.7a). Tangency along the generator QP means that the layer
normals of the two FCDs agree precisely where the FCDs come into contact with each other. Similarly, an FCD with bounding cone apex at P may be joined smoothly onto a family of concentric spheres centered at P (Fig. 4.7b) [115]. This is fortunate because concentric spheres, like FCDs, have a focal set of dimension less than two, avoiding energetically costly cusp wall defects.
(a) (b)
Figure 4.7: (a) The law of corresponding cones (LCC) illustrated for a pair of Type I focal conic domains (blue and red) with focal ellipses E1 and E2 tangent at point
Q, focal hyperbolaeH1 andH2 intersecting at point P, and bounding cones C1 and
C2 tangent along the generator QP. (b) A Type I focal conic domain (blue) with
elliptical focal curve E and bounding cone C (outlined by black lines) intersecting the hyperbolic focal curve (a straight line in this case) at pointP, with layers joined smoothly onto a family of spherical layers concentric about P.
What about Type II FCDs? Do they have a law of corresponding cones? For Type I FCDs, the key was to first construct a bounding cone consisting of generators (layer normals). Let’s starting by considering, for Type II FCDs, the cone of revolutionC with apex on the hyperbola atH~(v) and containing the elliptical focal curve. Unfortunately, the FCD generators no longer lie on the cone. However, if we continue the lines that
formed the cone, from E~(u) to H~(v) for u∈[0,2π), upward past H~(v), we form an inverted cone of revolution C0 thatdoes consist of generators of the Type II FCD. By
keeping only the region of the Type II FCD that is inside C0, we can join the Type II
FCD onto other Type II FCDs, onto concentric spheres, and even onto Type I FCDs, all with bounding cone apex or center at H~(v). This construction is illustrated in Fig. 4.8.
Figure 4.8: Left: Corresponding cones for Type II FCDs. Right: Composite texture of Type I and II FCDs and concentric spheres. Physical focal sets are shown in black, virtual focal sets in gray. For both figures, zero eccentricity is taken for simplicity.